Bloch-wave homogenization on large time scales and dispersive effective wave equations

Original · EN

We investigate second order linear wave equations in periodic media, aiming at the derivation of effective equations in ⁿ, n ∈ {1, 2, 3}. Standard homogenization theory provides, for the limit of a small periodicity length >0, an effective second order wave equation that describes solutions on time intervals [0,T]. In order to approximate solutions on large time intervals [0,T⁻²], one has to use a dispersive, higher order wave equation. In this work, we provide a well-posed, weakly dispersive effective equation, and an estimate for errors between the solution of the original heterogeneous problem and the solution of the dispersive wave equation. We use Bloch-wave analysis to identify a family of relevant limit models and introduce an approach to select a well-posed effective model under symmetry assumptions on the periodic structure. The analytical results are confirmed and illustrated by numerical tests.